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A [[Linear topology|linear topology]] of a ring <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010790/a0107901.png" /> in which the fundamental system of neighbourhoods of zero consists of the powers <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010790/a0107902.png" /> of some two-sided ideal <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010790/a0107903.png" />. The topology is then said to be <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010790/a0107904.png" />-adic, and the ideal <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010790/a0107905.png" /> is said to be the defining ideal of the topology. The closure of any set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010790/a0107906.png" /> in the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010790/a0107907.png" />-adic topology is equal to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010790/a0107908.png" />; in particular, the topology is separable if, and only if, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010790/a0107909.png" />. The separable completion <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010790/a01079010.png" /> of the ring <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010790/a01079011.png" /> in an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010790/a01079012.png" />-adic topology is isomorphic to the projective limit <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010790/a01079013.png" />.
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The <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010790/a01079014.png" />-adic topology of an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010790/a01079015.png" />-module <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010790/a01079016.png" /> is defined in a similar manner: its fundamental system of neighbourhoods of zero is given by the submodules <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010790/a01079017.png" />; in the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010790/a01079018.png" />-adic topology <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010790/a01079019.png" /> becomes a topological <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010790/a01079020.png" />-module.
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Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010790/a01079021.png" /> be a commutative ring with identity with an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010790/a01079022.png" />-adic topology and let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010790/a01079023.png" /> be its completion; if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010790/a01079024.png" /> is an ideal of finite type, the topology in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010790/a01079025.png" /> is <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010790/a01079026.png" />-adic, and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010790/a01079027.png" />. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010790/a01079028.png" /> is a maximal ideal, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010790/a01079029.png" /> is a local ring with maximal ideal <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010790/a01079030.png" />. A local ring topology is an adic topology defined by its maximal ideal (an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010790/a01079032.png" />-adic topology).
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A [[Linear topology|linear topology]] of a ring  $  A $
 +
in which the fundamental system of neighbourhoods of zero consists of the powers  $  \mathfrak A  ^ {n} $
 +
of some two-sided ideal  $  \mathfrak A $.  
 +
The topology is then said to be $  \mathfrak A $-
 +
adic, and the ideal  $  \mathfrak A $
 +
is said to be the defining ideal of the topology. The closure of any set  $  F \subset  A $
 +
in the  $  \mathfrak A $-
 +
adic topology is equal to  $  \cap _ {n \geq  0 }  ( F + \mathfrak A  ^ {n} ) $;
 +
in particular, the topology is separable if, and only if, $  \cap _ {n \geq  0 }  \mathfrak A  ^ {n} = (0) $.  
 +
The separable completion  $  \widehat{A}  $
 +
of the ring $  A $
 +
in an $  \mathfrak A $-
 +
adic topology is isomorphic to the projective limit  $  \lim\limits _  \leftarrow  ( A / \mathfrak A  ^ {n} ) $.
  
A fundamental tool in the study of adic topologies of rings is the Artin–Rees lemma: Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010790/a01079033.png" /> be a commutative Noetherian ring, let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010790/a01079034.png" /> be an ideal in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010790/a01079035.png" />, let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010790/a01079036.png" /> be an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010790/a01079037.png" />-module of finite type, and let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010790/a01079038.png" /> be a submodule of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010790/a01079039.png" />. Then there exists a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010790/a01079040.png" /> such that, for any <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010790/a01079041.png" />, the following equality is valid:
+
The  $  \mathfrak A $-
 +
adic topology of an  $  A $-
 +
module  $  M $
 +
is defined in a similar manner: its fundamental system of neighbourhoods of zero is given by the submodules  $  \mathfrak A  ^ {n} M $;
 +
in the  $  \mathfrak A $-
 +
adic topology  $  M $
 +
becomes a topological  $  A $-
 +
module.
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010790/a01079042.png" /></td> </tr></table>
+
Let  $  A $
 +
be a commutative ring with identity with an  $  \mathfrak A $-
 +
adic topology and let  $  \widehat{A}  $
 +
be its completion; if  $  \mathfrak A $
 +
is an ideal of finite type, the topology in  $  \widehat{A}  $
 +
is  $  \widehat{\mathfrak A}  $-
 +
adic, and  $  {\widehat{\mathfrak A}  } {}  ^ {n} = \mathfrak A  ^ {n} \widehat{A}  $.  
 +
If  $  \mathfrak A $
 +
is a maximal ideal, then  $  \widehat{A}  $
 +
is a local ring with maximal ideal  $  \widehat{\mathfrak A}  $.  
 +
A local ring topology is an adic topology defined by its maximal ideal (an  $  \mathfrak m $-
 +
adic topology).
  
The topological interpretation of the Artin–Rees lemma shows that the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010790/a01079043.png" />-adic topology of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010790/a01079044.png" /> is induced by the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010790/a01079045.png" />-adic topology of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010790/a01079046.png" />. It follows that the completion <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010790/a01079047.png" /> of a ring <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010790/a01079048.png" /> in the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010790/a01079049.png" />-adic topology is a flat <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010790/a01079050.png" />-module (cf. [[Flat module|Flat module]]), that the completion <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010790/a01079051.png" /> of the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010790/a01079052.png" />-module <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010790/a01079053.png" /> of finite type is identical with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010790/a01079054.png" />, and that Krull's theorem holds: The <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010790/a01079055.png" />-adic topology of a Noetherian ring is separable if and only if the set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010790/a01079056.png" /> contains no zero divisors. In particular, the topology is separable if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010790/a01079057.png" /> is contained in the (Jacobson) radical of the ring.
+
A fundamental tool in the study of adic topologies of rings is the Artin–Rees lemma: Let  $  A $
 +
be a commutative Noetherian ring, let  $  \mathfrak A $
 +
be an ideal in  $  A $,
 +
let  $  E $
 +
be an  $  A $-
 +
module of finite type, and let  $  F $
 +
be a submodule of  $  E $.
 +
Then there exists a  $  k $
 +
such that, for any  $  n \geq  0 $,
 +
the following equality is valid:
 +
 
 +
$$
 +
\mathfrak A  ^ {n} ( \mathfrak A  ^ {k} E \cap F )  = \
 +
\mathfrak A ^ {k + n } E \cap F .
 +
$$
 +
 
 +
The topological interpretation of the Artin–Rees lemma shows that the $  \mathfrak A $-
 +
adic topology of $  F $
 +
is induced by the $  \mathfrak A $-
 +
adic topology of $  E $.  
 +
It follows that the completion $  \widehat{A}  $
 +
of a ring $  A $
 +
in the $  \mathfrak A $-
 +
adic topology is a flat $  A $-
 +
module (cf. [[Flat module|Flat module]]), that the completion $  \widehat{E}  $
 +
of the $  A $-
 +
module $  E $
 +
of finite type is identical with $  E \otimes _ {A} \widehat{A}  $,  
 +
and that Krull's theorem holds: The $  \mathfrak A $-
 +
adic topology of a Noetherian ring is separable if and only if the set $  1 + \mathfrak A $
 +
contains no zero divisors. In particular, the topology is separable if $  \mathfrak A $
 +
is contained in the (Jacobson) radical of the ring.
  
 
====References====
 
====References====
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  O. Zariski,  P. Samuel,  "Commutative algebra" , '''2''' , Springer  (1975)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  N. Bourbaki,  "Elements of mathematics. Commutative algebra" , Addison-Wesley  (1972)  (Translated from French)</TD></TR></table>
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  O. Zariski,  P. Samuel,  "Commutative algebra" , '''2''' , Springer  (1975)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  N. Bourbaki,  "Elements of mathematics. Commutative algebra" , Addison-Wesley  (1972)  (Translated from French)</TD></TR></table>

Latest revision as of 16:09, 1 April 2020


A linear topology of a ring $ A $ in which the fundamental system of neighbourhoods of zero consists of the powers $ \mathfrak A ^ {n} $ of some two-sided ideal $ \mathfrak A $. The topology is then said to be $ \mathfrak A $- adic, and the ideal $ \mathfrak A $ is said to be the defining ideal of the topology. The closure of any set $ F \subset A $ in the $ \mathfrak A $- adic topology is equal to $ \cap _ {n \geq 0 } ( F + \mathfrak A ^ {n} ) $; in particular, the topology is separable if, and only if, $ \cap _ {n \geq 0 } \mathfrak A ^ {n} = (0) $. The separable completion $ \widehat{A} $ of the ring $ A $ in an $ \mathfrak A $- adic topology is isomorphic to the projective limit $ \lim\limits _ \leftarrow ( A / \mathfrak A ^ {n} ) $.

The $ \mathfrak A $- adic topology of an $ A $- module $ M $ is defined in a similar manner: its fundamental system of neighbourhoods of zero is given by the submodules $ \mathfrak A ^ {n} M $; in the $ \mathfrak A $- adic topology $ M $ becomes a topological $ A $- module.

Let $ A $ be a commutative ring with identity with an $ \mathfrak A $- adic topology and let $ \widehat{A} $ be its completion; if $ \mathfrak A $ is an ideal of finite type, the topology in $ \widehat{A} $ is $ \widehat{\mathfrak A} $- adic, and $ {\widehat{\mathfrak A} } {} ^ {n} = \mathfrak A ^ {n} \widehat{A} $. If $ \mathfrak A $ is a maximal ideal, then $ \widehat{A} $ is a local ring with maximal ideal $ \widehat{\mathfrak A} $. A local ring topology is an adic topology defined by its maximal ideal (an $ \mathfrak m $- adic topology).

A fundamental tool in the study of adic topologies of rings is the Artin–Rees lemma: Let $ A $ be a commutative Noetherian ring, let $ \mathfrak A $ be an ideal in $ A $, let $ E $ be an $ A $- module of finite type, and let $ F $ be a submodule of $ E $. Then there exists a $ k $ such that, for any $ n \geq 0 $, the following equality is valid:

$$ \mathfrak A ^ {n} ( \mathfrak A ^ {k} E \cap F ) = \ \mathfrak A ^ {k + n } E \cap F . $$

The topological interpretation of the Artin–Rees lemma shows that the $ \mathfrak A $- adic topology of $ F $ is induced by the $ \mathfrak A $- adic topology of $ E $. It follows that the completion $ \widehat{A} $ of a ring $ A $ in the $ \mathfrak A $- adic topology is a flat $ A $- module (cf. Flat module), that the completion $ \widehat{E} $ of the $ A $- module $ E $ of finite type is identical with $ E \otimes _ {A} \widehat{A} $, and that Krull's theorem holds: The $ \mathfrak A $- adic topology of a Noetherian ring is separable if and only if the set $ 1 + \mathfrak A $ contains no zero divisors. In particular, the topology is separable if $ \mathfrak A $ is contained in the (Jacobson) radical of the ring.

References

[1] O. Zariski, P. Samuel, "Commutative algebra" , 2 , Springer (1975)
[2] N. Bourbaki, "Elements of mathematics. Commutative algebra" , Addison-Wesley (1972) (Translated from French)
How to Cite This Entry:
Adic topology. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Adic_topology&oldid=45034
This article was adapted from an original article by V.I. Danilov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article